Robust solvers for symmetric positive definite operators and weighted poincaré inequalities

Authors:
Yalchin Efendiev;Juan Galvis;Raytcho Lazarov;Joerg Willems
Affiliations:
Dept. Mathematics, Texas A&M University, College Station, TX;Dept. Mathematics, Texas A&M University, College Station, TX;Dept. Mathematics, Texas A&M University, College Station, TX;RICAM, Linz, Austria
Venue:
LSSC'11 Proceedings of the 8th international conference on Large-Scale Scientific Computing
Year:
2011

Citing 5
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Abstract

An abstract setting for robustly preconditioning symmetric positive definite (SPD) operators is presented. The term "robust" refers to the property of the condition numbers of the preconditioned systems being independent of mesh parameters and problem parameters. Important instances of such problem parameters are in particular (highly varying) coefficients. The method belongs to the class of additive Schwarz preconditioners. The paper gives an overview of the results obtained in a recent paper by the authors. It, furthermore, focuses on the importance of weighted Poincaré inequalities, whose notion is extended to general SPD operators, for the analysis of stable decompositions. To demonstrate the applicability of the abstract preconditioner the scalar elliptic equation and the stream function formulation of Brinkman's equations in two spatial dimensions are considered. Several numerical examples are presented.